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<H1><A NAME="SECTION034110000000000000000"></A>
<A NAME="sec_GNEPErrorBounds"></A>
<BR>
Error Bounds for the Generalized Nonsymmetric Eigenproblem
</H1>

<P>
We start by stating the simplest error bounds for individual eigenvalues
and eigenvectors and leave the more complicated ones to subsequent
subsections. <A NAME="12054"></A>

<P>
As discussed in section&nbsp;<A HREF="node55.html#sec_gnep_comp">2.4.8</A> ,
from a computational point of view it is more natural to define
the generalized nonsymmetric eigenvalue problem in
the form 
<!-- MATH
 $\beta A x = \alpha B x$
 -->
<IMG
 WIDTH="97" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img714.gif"
 ALT="$\beta A x = \alpha B x$">
with 
<!-- MATH
 $\lambda = \alpha / \beta$
 -->
<IMG
 WIDTH="69" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img51.gif"
 ALT="$\lambda = \alpha/\beta$">
instead of 
<!-- MATH
 $A x = \lambda B x$
 -->
<IMG
 WIDTH="85" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="img176.gif"
 ALT="$Ax = \lambda Bx$">.
The eigenvalue <IMG
 WIDTH="15" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="img23.gif"
 ALT="$\lambda$">
is represented
as a pair 
<!-- MATH
 $(\alpha, \beta)$
 -->
<IMG
 WIDTH="48" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img181.gif"
 ALT="$(\alpha,\beta)$">,
where a finite
eigenvalue has <IMG
 WIDTH="47" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img715.gif"
 ALT="$\beta \neq 0$">
and an infinite eigenvalue has <IMG
 WIDTH="47" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img716.gif"
 ALT="$\beta = 0$">.
As in the standard nonsymmetric eigenvalue problem we have both right
and left eigenvectors <IMG
 WIDTH="46" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img717.gif"
 ALT="$x \neq 0$">
and <IMG
 WIDTH="46" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img718.gif"
 ALT="$y \neq 0$">,
respectively, defined as
<BR>
<DIV ALIGN="RIGHT">


<!-- MATH
 \begin{equation}
\beta A x = \alpha B x, \quad \beta y^H A  = \alpha y^H B .
\end{equation}
 -->

<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><A NAME="eq411.1"></A><IMG
 WIDTH="238" HEIGHT="30" BORDER="0"
 SRC="img719.gif"
 ALT="\begin{displaymath}
\beta A x = \alpha B x, \quad \beta y^H A = \alpha y^H B .
\end{displaymath}"></TD>
<TD WIDTH=10 ALIGN="RIGHT">
(4.8)</TD></TR>
</TABLE>
<BR CLEAR="ALL"></DIV><P></P>
Error bounds for eigenvalues are stated in terms of
the distance between pairs 
<!-- MATH
 $(\alpha, \beta)$
 -->
<IMG
 WIDTH="48" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img181.gif"
 ALT="$(\alpha,\beta)$">
and 
<!-- MATH
 $(\alpha',\beta')$
 -->
<IMG
 WIDTH="57" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img720.gif"
 ALT="$(\alpha',\beta')$">.
Let 
<!-- MATH
 $\lambda = \alpha/\beta$
 -->
<IMG
 WIDTH="69" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img51.gif"
 ALT="$\lambda = \alpha/\beta$">
and 
<!-- MATH
 $\lambda' = \alpha'/\beta'$
 -->
<IMG
 WIDTH="82" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img721.gif"
 ALT="$\lambda' = \alpha'/\beta'$">.
Then the <EM>chordal distance</EM><A NAME="12058"></A> between <IMG
 WIDTH="15" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="img23.gif"
 ALT="$\lambda$">
and
<IMG
 WIDTH="19" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="img722.gif"
 ALT="$\lambda'$">
(see section&nbsp;<A HREF="node99.html#secGSEPFurtherDetails">4.10.1</A>)
can equivalently be expressed as the chordal distance between two pairs:
<BR>
<DIV ALIGN="RIGHT">


<!-- MATH
 \begin{equation}
{\cal X}( \lambda,\lambda') =
{\cal X}((\alpha, \beta), (\alpha',\beta')) = \frac{|\alpha \beta' -
\beta \alpha' |}{\sqrt{|\alpha|^2 + |\beta|^2} \sqrt{|\alpha'|^2 + |\beta'|^2}}.
\end{equation}
 -->

<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><A NAME="eq411.2"></A><IMG
 WIDTH="442" HEIGHT="50" BORDER="0"
 SRC="img723.gif"
 ALT="\begin{displaymath}
{\cal X}( \lambda,\lambda') =
{\cal X}((\alpha, \beta), (\al...
...beta\vert^2} \sqrt{\vert\alpha'\vert^2 + \vert\beta'\vert^2}}.
\end{displaymath}"></TD>
<TD WIDTH=10 ALIGN="RIGHT">
(4.9)</TD></TR>
</TABLE>
<BR CLEAR="ALL"></DIV><P></P>

<P>
Now we state our error bounds.
Let 
<!-- MATH
 $({\alpha}_i, {\beta}_i), i = 1, \ldots, n$
 -->
<IMG
 WIDTH="152" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img724.gif"
 ALT="$({\alpha}_i, {\beta}_i), i = 1, \ldots, n$">
be the eigenvalues of <B>(<I>A</I>, <I>B</I>)</B>,
let <B><I>x</I><SUB><I>i</I></SUB></B> be a right eigenvector corresponding to 
<!-- MATH
 $({\alpha}_i, {\beta}_i)$
 -->
<IMG
 WIDTH="58" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img725.gif"
 ALT="$({\alpha}_i, {\beta}_i)$">:

<!-- MATH
 ${\beta}_i A x_i = {\alpha}_i B x_i$
 -->
<IMG
 WIDTH="117" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img726.gif"
 ALT="${\beta}_i A x_i = {\alpha}_i B x_i$">
and let

<!-- MATH
 $({\hat{\alpha}}_i, {\hat{\beta}}_i)$
 -->
<IMG
 WIDTH="58" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="img727.gif"
 ALT="$({\hat{\alpha}}_i, {\hat{\beta}}_i)$">
and <IMG
 WIDTH="20" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img356.gif"
 ALT="$\hat{x}_i$">
be the corresponding eigenvalues and
eigenvectors computed by the expert driver routine
xGGEVX (see subsection <A HREF="node35.html#sec_gnep_driver">2.3.5.2</A>).
<A NAME="12076"></A><A NAME="12077"></A><A NAME="12078"></A><A NAME="12079"></A>

<P>
The approximate error bounds<A NAME="footfnm 0"><SUP>4.10</SUP></A>for the computed eigenvalues are
<BR><P></P>
<DIV ALIGN="CENTER">

<!-- MATH
 \begin{displaymath}
{\cal X}(({\hat{\alpha}}_i, {\hat{\beta}}_i), ({\alpha}_i, {\beta}_i)) \leq
{\tt EERRBD}(i) .
\end{displaymath}
 -->


<IMG
 WIDTH="243" HEIGHT="31" BORDER="0"
 SRC="img728.gif"
 ALT="\begin{displaymath}
{\cal X}(({\hat{\alpha}}_i, {\hat{\beta}}_i), ({\alpha}_i, {\beta}_i)) \leq
{\tt EERRBD}(i) .
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
The approximate error bounds
<A NAME="12087"></A><A NAME="12088"></A>
for the computed eigenvectors <IMG
 WIDTH="20" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img356.gif"
 ALT="$\hat{x}_i$">,
which
bound the acute angles between the computed eigenvectors and the true
eigenvectors <B><I>x</I><SUB><I>i</I></SUB></B> are
<A NAME="12090"></A>
<A NAME="12091"></A>
<BR><P></P>
<DIV ALIGN="CENTER">

<!-- MATH
 \begin{displaymath}
\theta(\hat{x}_i, x_i) \leq {\tt VERRBD}(i) .
\end{displaymath}
 -->


<IMG
 WIDTH="162" HEIGHT="31" BORDER="0"
 SRC="img729.gif"
 ALT="\begin{displaymath}
\theta(\hat{x}_i, x_i) \leq {\tt VERRBD}(i) .
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
The same bounds also hold for the computed left eigenvectors.

<P>
These bounds can be computed by the following code fragment:

<P>
<PRE>
      EPSMCH = SLAMCH( 'E' )
*     Compute the generalized eigenvalues and eigenvectors of (A,B)
*     ALPHAR/BETA contains the real parts of the eigenvalues
*     ALPHAI/BETA contains the imaginary parts of the eigenvalues
*     VL contains the left eigenvectors
*     VR contains the right eigenvectors
      CALL SGGEVX( 'P', 'V', 'V', 'B', N, A, LDA, B, LDB, ALPHAR,
     $             ALPHAI, BETA, VL, LDVL, VR, LDVR, ILO, IHI, LSCALE,
     $             RSCALE, ABNRM, BBNRM, RCONDE, RCONDV, WORK, LWORK,
     $             IWORK, BWORK, INFO )
      IF( INFO.GT.0 ) THEN
         PRINT *,'INFO =', info, ' from SGGEVX.'
      ELSE IF( N.GT.0 ) THEN
         ABNORM = SLAPY2( ABNRM, BBNRM )
         DO 10 I = 1,N
            EERRBD(I) = EPSMCH*ABNORM/RCONDE(I)
            VERRBD(I) = EPSMCH*ABNORM/RCONDV(I)
   10    CONTINUE
      END IF
</PRE>

<P>
For example, suppose<A NAME="tex2html2431"
 HREF="footnode.html#foot12096"><SUP>4.1</SUP></A><code>SLAMCH('E')</code> 
<!-- MATH
 $= 2^{-24} = 5.960  \cdot 10^{-8}$
 -->
<IMG
 WIDTH="167" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="img730.gif"
 ALT="$ = 2^{-24} = 5.960 \cdot 10^{-8}$">
and
<BR><P></P>
<DIV ALIGN="CENTER">

<!-- MATH
 \begin{displaymath}
A = \left( \begin{array}{cccc}
           -132~~   &   -88~~  &    84    &    104     \\
           -158.4   &   -79.2  &  ~~76.8  &  ~~129.6   \\
          ~~129.6   &  ~~81.6  &   -79.2  &   -100.8   \\
            160     &    84    &   -80~~  &   -132~~   \\
           \end{array} \right )
\end{displaymath}
 -->


<IMG
 WIDTH="319" HEIGHT="93" BORDER="0"
 SRC="img731.gif"
 ALT="\begin{displaymath}
A = \left( \begin{array}{cccc}
-132~~ &amp; -88~~ &amp; 84 &amp; 104 \\...
...100.8 \\
160 &amp; 84 &amp; -80~~ &amp; -132~~ \\
\end{array} \right )
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
and
<BR><P></P>
<DIV ALIGN="CENTER">

<!-- MATH
 \begin{displaymath}
B = \left( \begin{array}{cccc}
           -60~~  &  -50~~  &   40   &   50   \\
           -69~~  &  -46.4  &   38   & ~~58.2 \\
          ~~58.8  &   46    &  -37.6 &  -48~~ \\
            70    &   50    &  -40~~ &  -60~~ \\
           \end{array} \right ).
\end{displaymath}
 -->


<IMG
 WIDTH="310" HEIGHT="93" BORDER="0"
 SRC="img732.gif"
 ALT="\begin{displaymath}
B = \left( \begin{array}{cccc}
-60~~ &amp; -50~~ &amp; 40 &amp; 50 \\
...
... -48~~ \\
70 &amp; 50 &amp; -40~~ &amp; -60~~ \\
\end{array} \right ).
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>

<P>
For this problem, the exact eigenvalues, eigenvectors, and eigenvalue
condition numbers are known.  Then the true eigenvalues, computed eigenvalues,
approximate error bounds, and true error bounds are
given in the following table.

<P>
<DIV ALIGN="CENTER">
<TABLE CELLPADDING=3 BORDER="1">
<TR><TD ALIGN="CENTER"><B><I>i</I></B></TD>
<TD ALIGN="CENTER">1</TD>
<TD ALIGN="CENTER">2</TD>
<TD ALIGN="CENTER">3</TD>
<TD ALIGN="CENTER">4</TD>
</TR>
<TR><TD ALIGN="CENTER"><IMG
 WIDTH="20" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img523.gif"
 ALT="$\lambda_i$"></TD>
<TD ALIGN="CENTER"><B>1</B></TD>
<TD ALIGN="CENTER"><B>2</B></TD>
<TD ALIGN="CENTER"><B>3</B></TD>
<TD ALIGN="CENTER"><B>4</B></TD>
</TR>
<TR><TD ALIGN="CENTER">
<!-- MATH
 $\hat{\lambda}_i$
 -->
<IMG
 WIDTH="20" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="img530.gif"
 ALT="$\hat{\lambda}_i$"></TD>
<TD ALIGN="CENTER">
<!-- MATH
 $1.0000529 \cdot 10^0~~$
 -->
<IMG
 WIDTH="128" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="img733.gif"
 ALT="$1.0000529 \cdot 10^0~~$"></TD>
<TD ALIGN="CENTER">
<!-- MATH
 $1.9999847 \cdot 10^0~~$
 -->
<IMG
 WIDTH="128" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="img734.gif"
 ALT="$1.9999847 \cdot 10^0~~$"></TD>
<TD ALIGN="CENTER">
<!-- MATH
 $2.9999785 \cdot 10^0~~$
 -->
<IMG
 WIDTH="128" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="img735.gif"
 ALT="$2.9999785 \cdot 10^0~~$"></TD>
<TD ALIGN="CENTER">
<!-- MATH
 $4.0002117 \cdot 10^0~~$
 -->
<IMG
 WIDTH="128" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="img736.gif"
 ALT="$4.0002117 \cdot 10^0~~$"></TD>
</TR>
<TR><TD ALIGN="CENTER"><code>EERRBD</code><B>(<I>i</I>)</B></TD>
<TD ALIGN="CENTER">
<!-- MATH
 $9.4562565 \cdot 10^{-5}$
 -->
<IMG
 WIDTH="127" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="img737.gif"
 ALT="$9.4562565 \cdot 10^{-5}$"></TD>
<TD ALIGN="CENTER">
<!-- MATH
 $5.3651773 \cdot 10^{-5}$
 -->
<IMG
 WIDTH="127" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="img738.gif"
 ALT="$5.3651773 \cdot 10^{-5}$"></TD>
<TD ALIGN="CENTER">
<!-- MATH
 $5.6895351 \cdot 10^{-5}$
 -->
<IMG
 WIDTH="127" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="img739.gif"
 ALT="$5.6895351 \cdot 10^{-5}$"></TD>
<TD ALIGN="CENTER">
<!-- MATH
 $1.2976544 \cdot 10^{-4}$
 -->
<IMG
 WIDTH="127" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="img740.gif"
 ALT="$1.2976544 \cdot 10^{-4}$"></TD>
</TR>
<TR><TD ALIGN="CENTER">
<!-- MATH
 ${\cal X}(\hat{\lambda}_i, \lambda_i )$
 -->
<IMG
 WIDTH="72" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="img741.gif"
 ALT="${\cal X}(\hat{\lambda}_i, \lambda_i )$"></TD>
<TD ALIGN="CENTER">
<!-- MATH
 $3.6398560 \cdot 10^{-5}$
 -->
<IMG
 WIDTH="127" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="img742.gif"
 ALT="$3.6398560 \cdot 10^{-5}$"></TD>
<TD ALIGN="CENTER">
<!-- MATH
 $2.3351108 \cdot 10^{-5}$
 -->
<IMG
 WIDTH="127" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="img743.gif"
 ALT="$2.3351108 \cdot 10^{-5}$"></TD>
<TD ALIGN="CENTER">
<!-- MATH
 $2.2801253 \cdot 10^{-6}$
 -->
<IMG
 WIDTH="127" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="img744.gif"
 ALT="$2.2801253 \cdot 10^{-6}$"></TD>
<TD ALIGN="CENTER">
<!-- MATH
 $1.0059956 \cdot 10^{-5}$
 -->
<IMG
 WIDTH="127" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="img745.gif"
 ALT="$1.0059956 \cdot 10^{-5}$"></TD>
</TR>
<TR><TD ALIGN="CENTER"><code>VERRBD</code><B>(<I>i</I>)</B></TD>
<TD ALIGN="CENTER">
<!-- MATH
 $1.4116328 \cdot 10^{-4}$
 -->
<IMG
 WIDTH="127" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="img746.gif"
 ALT="$1.4116328 \cdot 10^{-4}$"></TD>
<TD ALIGN="CENTER">
<!-- MATH
 $1.4498082 \cdot 10^{-4}$
 -->
<IMG
 WIDTH="127" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="img747.gif"
 ALT="$1.4498082 \cdot 10^{-4}$"></TD>
<TD ALIGN="CENTER">
<!-- MATH
 $6.8483077 \cdot 10^{-4}$
 -->
<IMG
 WIDTH="127" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="img748.gif"
 ALT="$6.8483077 \cdot 10^{-4}$"></TD>
<TD ALIGN="CENTER">
<!-- MATH
 $5.5552053 \cdot 10^{-4}$
 -->
<IMG
 WIDTH="127" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="img749.gif"
 ALT="$5.5552053 \cdot 10^{-4}$"></TD>
</TR>
<TR><TD ALIGN="CENTER">
<!-- MATH
 $\theta(\hat{l}_i,l_i)$
 -->
<IMG
 WIDTH="56" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="img750.gif"
 ALT="$\theta(\hat{l}_i,l_i)$"></TD>
<TD ALIGN="CENTER">
<!-- MATH
 $1.0050300 \cdot 10^{-5}$
 -->
<IMG
 WIDTH="127" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="img751.gif"
 ALT="$1.0050300 \cdot 10^{-5}$"></TD>
<TD ALIGN="CENTER">
<!-- MATH
 $6.9755580 \cdot 10^{-6}$
 -->
<IMG
 WIDTH="127" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="img752.gif"
 ALT="$6.9755580 \cdot 10^{-6}$"></TD>
<TD ALIGN="CENTER">
<!-- MATH
 $1.3587955 \cdot 10^{-5}$
 -->
<IMG
 WIDTH="127" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="img753.gif"
 ALT="$1.3587955 \cdot 10^{-5}$"></TD>
<TD ALIGN="CENTER">
<!-- MATH
 $4.2988235 \cdot 10^{-6}$
 -->
<IMG
 WIDTH="127" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="img754.gif"
 ALT="$4.2988235 \cdot 10^{-6}$"></TD>
</TR>
<TR><TD ALIGN="CENTER">
<!-- MATH
 $\theta(\hat{r}_i,r_i)$
 -->
<IMG
 WIDTH="61" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img755.gif"
 ALT="$\theta(\hat{r}_i,r_i)$"></TD>
<TD ALIGN="CENTER">
<!-- MATH
 $5.2165419 \cdot 10^{-5}$
 -->
<IMG
 WIDTH="127" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="img756.gif"
 ALT="$5.2165419 \cdot 10^{-5}$"></TD>
<TD ALIGN="CENTER">
<!-- MATH
 $1.4475762 \cdot 10^{-5}$
 -->
<IMG
 WIDTH="127" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="img757.gif"
 ALT="$1.4475762 \cdot 10^{-5}$"></TD>
<TD ALIGN="CENTER">
<!-- MATH
 $5.1648690 \cdot 10^{-5}$
 -->
<IMG
 WIDTH="127" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="img758.gif"
 ALT="$5.1648690 \cdot 10^{-5}$"></TD>
<TD ALIGN="CENTER">
<!-- MATH
 $7.9673846 \cdot 10^{-5}$
 -->
<IMG
 WIDTH="127" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="img759.gif"
 ALT="$7.9673846 \cdot 10^{-5}$"></TD>
</TR>
</TABLE>
</DIV>

<P>
<BR><HR>
<!--Table of Child-Links-->
<A NAME="CHILD_LINKS"></A>

<UL>
<LI><A NAME="tex2html5593"
 HREF="node101.html">Further Details: Error Bounds for the Generalized
Nonsymmetric Eigenproblem</A>
<UL>
<LI><A NAME="tex2html5594"
 HREF="node102.html">Overview</A>
<LI><A NAME="tex2html5595"
 HREF="node103.html">Balancing and Conditioning</A>
<LI><A NAME="tex2html5596"
 HREF="node104.html">Computing <B><I>s</I><SUB><I>i</I></SUB></B>, <IMG
 WIDTH="19" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img4.gif"
 ALT="$l_{\cal I}$">,
<IMG
 WIDTH="21" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="img5.gif"
 ALT="$r_{\cal I}$">
and
<IMG
 WIDTH="38" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img6.gif"
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<LI><A NAME="tex2html5597"
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<B> Next:</B> <A NAME="tex2html5592"
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<ADDRESS>
<I>Susan Blackford</I>
<BR><I>1999-10-01</I>
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